Read the given statements and conclusions carefully. Assuming that the information given in the statements is true, even if it appears to be at variance with commonly known facts, decide which of the given conclusions logically follow(s) from the statements. Statements: All polygons are angles. All angles are diagonals. All cones are cubes. All cubes are decagons. No diagonal is a cube. Conclusions: I. Some diagonals are polygons. II. All diagonals are decagons. III. No polygon is a cone. IV. Some cubes are angles.
Both conclusions I and III follow
Understanding the Statements and Conclusions Logic
This question requires us to analyze a set of statements and determine which of the given conclusions logically follow from them, assuming the statements are true.
Analyzing the Given Statements
Let's break down the statements provided:
- Statement 1: All polygons are angles. (Polygon → Angle)
- Statement 2: All angles are diagonals. (Angle → Diagonal)
- Statement 3: All cones are cubes. (Cone → Cube)
- Statement 4: All cubes are decagons. (Cube → Decagon)
- Statement 5: No diagonal is a cube. (Diagonal ↔ No Cube)
Combining Statements for Logical Deduction
We can combine some statements to find indirect relationships:
- From Statement 1 and 2: All polygons are angles, and all angles are diagonals. This means All polygons are diagonals. (Polygon → Angle → Diagonal)
- From Statement 3 and 4: All cones are cubes, and all cubes are decagons. This means All cones are decagons. (Cone → Cube → Decagon)
- Now consider Statement 5: No diagonal is a cube.
- Since All polygons are diagonals and No diagonal is a cube, it follows that No polygon is a cube.
- Since All cones are cubes and No diagonal is a cube, it follows that No cone is a diagonal.
- Since All cubes are decagons and No diagonal is a cube, it implies the set of diagonals and the set of cubes (and thus decagons related through cubes) are separate.
Evaluating the Conclusions
Let's examine each conclusion based on the deductions from the statements:
- Conclusion I: Some diagonals are polygons.
We deduced that All polygons are diagonals. If all of entity A are entity B, then it is logically true that some of entity B are entity A. Therefore, if All polygons are diagonals, then Some diagonals are polygons. This conclusion follows.
- Conclusion II: All diagonals are decagons.
We know that No diagonal is a cube and All cubes are decagons. This establishes a separation between diagonals and the set of cubes/decagons. There is no information suggesting that all diagonals must also be decagons. This conclusion does not follow.
- Conclusion III: No polygon is a cone.
We established that All polygons are diagonals and No diagonal is a cube. This implies No polygon is a cube. We are also given that All cones are cubes. If no polygon is a cube, and all cones are cubes, then certainly no polygon can be a cone. This conclusion follows.
- Conclusion IV: Some cubes are angles.
We know that All angles are diagonals and No diagonal is a cube. This means that the set of angles and the set of cubes are disjoint (they have no elements in common). Therefore, it is impossible for some cubes to be angles. This conclusion does not follow.
Summary of Conclusions
Based on our analysis:
- Conclusion I: Some diagonals are polygons. (Follows)
- Conclusion II: All diagonals are decagons. (Does not follow)
- Conclusion III: No polygon is a cone. (Follows)
- Conclusion IV: Some cubes are angles. (Does not follow)
Therefore, both conclusions I and III follow from the given statements.
Revision Table: Statements and Inferences
| Statements | Inference |
|---|---|
| All Polygons are Angles | Polygon → Angle |
| All Angles are Diagonals | Angle → Diagonal |
| All Polygons are Angles, All Angles are Diagonals | ⇒ All Polygons are Diagonals (Polygon → Diagonal) |
| All Cones are Cubes | Cone → Cube |
| All Cubes are Decagons | Cube → Decagon |
| All Cones are Cubes, All Cubes are Decagons | ⇒ All Cones are Decagons (Cone → Decagon) |
| No Diagonal is a Cube | Diagonal ↔ No Cube (Mutually exclusive sets) |
| All Polygons are Diagonals, No Diagonal is a Cube | ⇒ No Polygon is a Cube |
| All Cones are Cubes, No Diagonal is a Cube | ⇒ No Cone is a Diagonal |
| All Angles are Diagonals, No Diagonal is a Cube | ⇒ No Angle is a Cube |
Additional Information: Syllogism Rules
This question is based on principles of logical syllogisms, a form of deductive reasoning. Here are some basic rules applied here:
- All A are B: This means the set of A is a subset of the set of B. From this, it logically follows that Some B are A. It does *not* mean All B are A.
- No A is B: This means the set of A and the set of B are disjoint (they have no elements in common). From this, it logically follows that No B is A.
- Transitivity (Chain Rule): If All A are B and All B are C, then it logically follows that All A are C. (A → B → C ⇒ A → C)
- Negative Chain Rule: If All A are B and No B is C, then it logically follows that No A is C. (A → B, B ↔ No C ⇒ A ↔ No C) Similarly, if No A is B and All B are C, it does not necessarily mean No A is C (A could overlap with C if B is a subset of C). However, if All B are C and No A is B, you cannot say anything definitive about the relationship between A and C without more information. In this problem, we used the rule: All A are B, and No B is C ⇒ No A is C (e.g., Polygons are Diagonals, No Diagonal is Cube ⇒ No Polygon is Cube).
Applying these fundamental rules helps in correctly evaluating whether a conclusion logically follows from the given statements.
Similar Questions
Three statements are given, followed by three conclusions numbered I, II and III. Assuming the statements to be true, even if they seem to be at variance with commonly known facts, decide which of the conclusions logically follow(s) from the statements.
Statements:
All reds are pinks.
All whites are pinks.
All pinks are oranges.
Conclusions:
I. All whites are oranges.
II. All oranges are reds.
III. Some oranges are whites.
In the following question below are given some statements followed by some conclusions based on those statements. Taking the given statements to be true even if they seem to be at variance from commonly known facts. Read all the conclusions and then decide which of the given conclusion logically follows the given statements.
Statements:
I. All M are A.
II. No A is R.
Conclusion:
I. No M is R.
II. Some A are M.
III. Some R are A.
Three statements are given, followed by two conclusions numbered I and II. Assuming the statements to be true, even if they seem to be at variance with commonly known facts, decide which of the conclusions logically follow(s) from the statements.
Statements:
All c are f.
No f is t.
All p are c.
Conclusions:
I. Some t are c.
II. No p is t.
Three statements are followed by three conclusions numbered I, II and III. You have to consider these statements to be true, even if they seem to be at variance with commonly known facts. Decide which of the given conclusions logically follow(s) from the given statements.
Statements:
All peacocks are pigeons.
Some pigeons are pinks.
Some pinks are birds.
Conclusions:
(I) Some pinks are peacocks.
(II) Some birds are pigeons.
(III) Some peacocks are birds.
Three Statements are given followed by Three conclusions numbered I, II and III. Assuming the statements to be true, even if they seem to be at variance with commonly known facts, decide which of the conclusions logically follow(s) from the statements.
Statements:
All bottles are containers.
Some containers are buckets.
All buckets are soaps.
Conclusions:
I. Some soaps are bottles.
II. Some buckets are bottles.
III. Some soaps are containers.
In this question, three statements are given, followed by three conclusions numbered I, II and III. Assuming the statements to be true, even if they seem to be at variance with commonly known facts, decide which of the conclusions logically follows/follow from the statements.
Statements:
Some roses are yellow.
Some yellow are buses.
All buses are green.
Conclusions:
I. All yellow are roses.
II. Some buses are yellow.
III. All yellow are green.
Read the following statement carefully and identify the conclusion that follows.
Statement:
All plants are living beings. Some living beings are animals.Conclusions:
I. Some plants are animals.
II. All animals are plants.Three statements are given followed by three conclusions numbered I, Il and lII. Assuming the statements to be true, even if they seem to be at variance with commonly known facts, decide which of the conclusions logically follow(s) from the statements.
Statements:
All fruits are wines.
Some wines are juices.
All juices are alcohols.
Conclusions:
I. Some alcohols are juices.
Il. Some wines are alcohols.
III. Some wines are fruits.
In the following question below are given some statements followed by some conclusions based on those statements. Taking the given statements to be true even if they seem to be at variance from commonly known facts. Read all the conclusions and then decide which of the given conclusion logically follows the given statements.
Statements:
I. All R are M.
II. All L are M.
Conclusions:
I. Some M are not R.
II. Some L are R.
III. Some M are L.
Read the given statements and conclusions carefully. Assuming that the information given in the statements is true, even if it appears to be at variance with commonly known facts, decide which of the given conclusions logically follow(s) from the statements.
Statements:
Some laptops are phones.
Some phones are capsules.
Some capsules are pens.
Conclusions:
I. Some laptops are capsules.
II. Some phones are pens.
III. Some pens are capsules.
IV. Some pens are phones.
Important Questions from Conventional Syllogism
In this question, three statements followed by two conclusions numbered I and II have been given. You have to take the given statements to be true even if they seem to be at variance from the commonly facts and then decide which of the given conclusions logically follows from the given statements disregarding commonly known facts.
Statements: Some flats are apartments.
No apartment is a hall.
Some halls are rooms.
Conclusions: I. At least some rooms are flats.
II. No apartment is a room.
Read the given statements and conclusions carefully. Assuming that the information given in the statements is true, even if it appears to be at variance with commonly known facts, decide which of the given conclusions logically follow(s) from the statements.
Statements:
I. Some blue are red.
II. Some green are red.
Conclusions:
I. No blue is green.
II. No red is green.
In the question two statements are given, followed by three conclusions, I, II and III. You have to consider the statements to be true even if it seems to be at variance from commonly known facts. You have to decide which of the given conclusions, if any, follows from the given statements.
Statement 1 : Some cars are scooters.
Statement 2 : All scooters are buses.
Conclusion I : Some scooters are cars.
Conclusion II : Some buses are cars.
Conclusion III : All cars are buses.
Read the given statements and conclusions carefully. Assuming that the information in the statement is true even if they appear to be at variance with commonly known facts, choose the conclusion that logically follows from the statement beyond reasonable doubt.
Statement:-
1. Some dogs are cats.
2. All cats are birds.
Conclusion:-
1. Some dogs are birds.
2. No cat is dog.
In each question below are given three statements followed by two conclusions numbered I and II. You have to take the two/three given statements to be true even if they seem to be at variance from commonly known facts and then decide which of the given conclusions logically follows from the given statements disregarding commonly known facts.
Statement:
Some mobiles are laptops.
No laptop is computer.
All phones are computer.
Conclusion:
I. At least some mobiles are computer.
II. At least some phone are laptops.
